Publicacions Matemátiques, Vol 32 (1988, 115-121 .

ON EQUIVARIANT DEFORMATION OF MAPS ANTONIO VIDAL

Abstract

We work in the smooth category : manifolds and maps are meant to be smooth . Let G be a finite group acting on a connected closed manifold X and f an equivariant self-map on X with flA fixpointfree, where A is a closed invariant submanifold of X with codim A >- 3. The purpose of this paper is to give a proof using obstruction theory of the following fact : If X is simply connected and the action of G on X - A is free, then f is equivariantly deformable rel. A to fixed point free map if and only if the usual Lefschetz number L(fl (x A» = 0. As a consequence we obtain a special case of a theorem of Wilczynski ( cf . [12, Theorem A] ~ . Finally, motivated by Wilczynski's paper we present an interesting question concerning the equivariant version of the converse of the Lefschetz fixed point theorem.

1 . Preliminaries We recall here how the classical obstruction theory of deformations of mappings (cf. [1]) can be translated to the equivariant case (cf. [3]) . Let G be a finite group, (Y, B) a pair of G-spaces and (K, L) a given pair of finite regular G-complexes (cf . [4, p . 116] for the definition) . A G-map h : (K, L) -> (Y, B) is said to be equivariantly deformable into B if ther exists a G-homotopy ht : (K, L) -> (Y, B), t E I = [0,1], such that ho = h and hl (K) C B. We say that h is equivariantly q-deformable into B, if the partial map h(Y',L) is equivariantly deformable into B, K := K9 U L, K9 = qskeleton of K. Note that for regular G-complexes K we have that if g E G leaves any point x E X fixed, then g must leave the smallest subcomplex of K containing x pointwise fixed. Therefore, by an easy inductive argument on the skeletons of K we can show that K has the equivariant extension property with respect to L. This allows us to assume that ht (x) = x for every x E L and t E I in the definition of deformability. Let us assume that Bx 7~ 0 for all the subgroups H of G that occur as isotropy groups of simplexes in K, and both BH and Ya are pathwise connected . Then it is clear that h must be equivariantly O-deformable into B . The author would like to thank Professors Fadell and Kwasik for helpful suggestions

A . VIDAL

116

Moreover, if the inclusion BH C YH induces an epimorphism of fundamental groups gr1 (B H , y) ) 7r 1 (YH, y) for all y E B, then it can be proved easily, as in the non equivariant case, that every map h is equivariantly 1-deformable into B . Next, let q > 2 and assume that (YH, BH ) is q-simple for all subgroups H of G. Then the relative homotopy groups 7rq (YH, BH ) are abelian and can be used to define the following generic coefficient system for G, wq (G/H) := Irq (Y H ,B") (cf.[3]) . Now suppose that h(K -1 ) C B. For each orientable qsimplex a in K-L define an element d'G (h) (G) E 7rq (Y G ° , BG ° ),G,, the isotropy group of a, to be the element determined by the map h : (u, á) -) (YG°, BG°) . Since this last pair is simple it does not matter how the base points are choosen in the definition of d'G (h) . We can prove that de (h) is a relative equivariant -1 coclycle of K modulo L and dG (h) = 0 if and only if h rel . K equivariantly q-deformable into B. Now consider the n-dimensional obstruction set OG (h) to the equivariant deformation of the map h. It is a subset of the Bredon cohomology group HG (K, L; wq ) (cf. [3]) and is defined as in the classical case. We have that OG (h) :7É 0 if h is equivariantly q -1 deformable into B and if h is equivariantly q-deformable into B, then OG (h) contains the zero element . Next we wish to indicate how the converse of this latter statement can be proved . Let h be equivariantly q-1 deformable into B, h o , h 1 : (K, L) ) (Y, B) two -1) equivariant maps which satisfy the conditions h; (K C B, h; - h, i = 0, 1, a

G

and ho1V9-a = h1,K,-2 . Then it is possible to associate to any equivariant homotopy ht between ho and h1 which has the property that ht (x) = ho (x) = h1 (x) for all x E K-2 an equivariant separation cochain dG 1 (ht) E (K, L; wq ) satisfying the equation

1

6dG (ht)=dG(h1)-dG(ho) .

This separation cochain is defined as follows : Let há be the restriction of ht to KH . By the classical theory applied to this map (cf. [1]), there exists a cochain dq -1 (hH) E Cq -1 (KH,LH ;wq(G1H)) with 6dq -1 (h H) = dq(hH) - dq (hH ) .

Because we are dealing with equivariant maps we have (D. dq-1 (hH ') (°) _ dq-1(hH(gU)) for g : GlH -> GIH' a morphism in the orbit category and

a a q - 1 simplex in K. Then put dG 1 (ht ) (a) := dq-1(hG °) (a) and we have dq 1 (ht) E

1 CG (K, L; wq ) satisfying the equation (1) .

Furthermore, if d is an element of the cochain group CG 1 (K, L; wq ) and h is any map satisfying the condition h(K -1 ) C B, then it can be defined an equiv2 ariant deformation ht of h such that ht (x) = h(x) for all x E K - , h1 (Kq-1) C dq-1 (h t ) = d (cf. [3]) . Finally, by making use of this separation cochain B and we can easily prove the following :

DEFORMATION OF MAPS

1.17

1 .1 . Proposition . The G-map h is equivariantly q-deformable into B if and only if o E OG (h) . Suppose now that (Y, B) is a G-locally trivial pair over K; that means we have a G-map p : Y -> K with p(B) = K and for each x E K there is a G,,-invariant neighborhood U of x and G,,-homeomorphism of pairs qíu : (p -l U, P l U)

(U x P- ' (x), U x P-' (x))

such that pl@v (x ) y) = x for all (x, y) E U x p-1(x) and po := pea . It is known that a G-locally pair over a G-complex is an. equivariant (Serre) fibration (cf . [2]) . Therefore we have the equivariant version of Lemma 5.1 in [5] and consequently the following 1.2 . Proposition . If h is a cross section of the G-locally trivial pair (Y, B) over K, and h is equivariantly deformable into B then it is also in the family of cross sections. 2 . Application to selfmaps on manifolds Throughout we assume maps and manifolds are smooth . Let G be a finite group acting on a pathwise and simply connected closed manifold X. Let f : (X, A) -+ (X, A) be a G-map where A is closed invariant submanifold of X and fIA is fixed point free. In order to apply the preceeding material to this situation let us assume that the manifolds XH are pathwise and simple connected for all subgroups H of G. Choose an equivariant triangulation of X with A a G-invariant subcomplex (cf. [7, p. 216]) . Considered locally at x E X, the G-manifold X looks like a G,,-representation in the tangent space TZ X (cf. [8]) . Consequently, if A is the diagonal, it can easily be proved that the pair k := (X x X, X x X - A) with p : X x X --> X the projection on the first factor is a G-locally trivial pair over the G-complex X. By Proposition 1.2 the G-map f is equivariantly deformable to a fixed point free map if and only if the cross section (1, f) : X + X x X, (1, f) (x) :_ (x, f(x)), is equivariantly deformable into X x X- A . Finally, by repeated application of Propósition 1.1 for (K, L) := (X, A), (Y, B) := (X x X, X x X - A) and h := (1; f), we obtain the following result: 2.1. Theorem . The G-map f : (X, A) --> (X, A), with flA fixpointfree, is equivariantly deformable rel . A to fixed point free map if and only if o E Oc (f) C HG (X, A; (D), where n =dim X and w is the coefcient system given by w (G1H) := ir (XH). To simplify the notation we have written OG (f) instead of Oc ((1, f» . Note that o E Oc (f ) if and only if o E Oc (f ) for all q, 2 < q < n. For a given

118

A . VIDAL

equivariant triangulation of (X, A), only the q-simplexes a in X - A with q =dim Xc ° contribute in a non trivial manner to the obstruction set OG ( f ) . This is because for q _ 3. Our next step is to prove that in this special case the group Hc (X, A; w,y ) is torsion free. Let U be a closed invariant tubular neighborhood of A equivariantly diffeomorphic to an E > 0 disc bundle De in the normal bundle N(A, X) to A in X, via h : U -+ D (cf. [4, p.3061) . Then A is an equivariant strong deformation retract of U, and the groups HG (X, A; ¿un) and Hc (X, U; W n) are isomorphic. 0

0

0

Let V := h_ 1(D,/z) . Clearly V C U, and the inclusion (X-V, U-V) C (X, U) 0

0

induces an isomorphism HG (X, U ;Wn) - HG (X - V, U - V; CJn ) . Consider now the following equivariant deformation 0 t : X - A --> X - A of the identity on X - A. For x E X - A we define 0

if XEX-U h-l pth(x)

if

xE U

pt : D, - N0 ---+ De - N0 and N0 =the zero cross section is defined by v) := (x, (1 '- t)v + tEv/ilvli) . On N(A, X) we have an ortogonal metric,

where pt (x,

hence the maps

are G-maps and, consequently, the maps 0 t are equivariant 0 0 0 0 diffeomorphisms . However 0 1 (X-V) = X-U and 0 1 (U-V) = U-U, hence 0 1 pt

0

0

0

0

induces an isomorphism 0i : HG (X - V, U - V; Wj --> HG (X - U, U - U ; CJ n ), where (X - U, U - U) is an n-dimensional compact manifold with boundary on which G acts freely. Furthermore if the codim A >_ 3, then we have that path componente of X - A are simply connected, and consequently the manifold 0

X - U is aleo simply connected . Finally, put (Y,

ay) :=

0

0

(X - U, U - U) .

2 .2. Lemma. With the previous notation and taking the orientation induced on Y by that on X, we have that the group HG (Y, aY ; wn) is torsion free . ProoL First we note that the group Hc (Y, ay ; Wj is, by definition, equal to ; 7rn (X)) where Ir n (X) - z have the right Z G-module the group H" (Y/ G, ay/G structure given by the coefficient system Wn . Second, and this is essential for

DEFORMATION OF MAPS

11 9

the proof, we use the fact that if we consider the orientation on .Y induced by that on X, then the G-module structure on Z coincides with the orientation homomorphism w : 7rl (YIG) --> Z 2 for the manifold Y1G, that is w," (g) = 1z if w(g) = 1 and w (g) = -1 a if w (g) = -1. By Theorem 2.1 in [11], (Y, ay) is a Poincaré pair of formal dimension n. Then Lemma 1.2 in [10] yields that the cup product with the fundamental class [Y/G, 8Y/G]n : H" (Y/G, 8Y/G; Z) -> Ha (Y/G; Z') is an isomorphism, where Z is considered with the Z G-right module structure given by_ w" and Z t is the left ZG-module defined as follows . For A = En(g)g E ZG let a := En(g)w(g)g-1 and define gz := zg = w(g)w" (g)(z) . However with the above choice w (g)wn (g) is always the identity and Z a is trivial as Z G-left module . Therefore Ho (Y1G; Z') is a free abelain group with as many generators as there are path components in Y/G . This proves the lemma. Now consider the homomorphism induced in cohomology by the inclusion CG (X, A; w") -> C* (X, A; Z),

ce : HG (X, A; w" ) ( ---) H* (X, A ; Z) : Q . In the opposite direction, let /1 be the transfer. At the cochain level, it is defined as follows : OZ(o') := EgCG w (g)z(g-IQ), z E Z" (X, A; 7r" (X)) . We have that the composition fa = -1GI, multiplication with IGI, the order of the group G . From this equation and the fact that the group HG (X, A; w") has no torsion, it follows that a must be monomorphism . On the other hand a maps the n-dimensional obstruction set 0'G (f) into the classical one . This last set has been computed by Fadell in [5] . It has only one element, the Lefschetz class of f(X,A) with coeficients in Z, and this is zero if and only if the usual Lefschetz number L(f)(X,A1) = 0. In summary we have the following : 2 .3. Theorem . Assuming in addition to the hypothesis of Theorem 2.1 that G aets freely on X-A and codim A >_ 3, we have that the G-map f : (X, A) > (X, A) is equivariantly deformable rel. A to a fized point free map if and only °f L(fl(x,Al) = 0-

2 .4 . Corollary . Suppose G aets semifreely on X with X' a pathwise and simply conneeted manifold of codim XG >_ 3. Then a G-map f : X -> X is equivariantly deformable to a fized point free map if and only if the Lefschetz numbbes L(f) = L(f G) = 0. 2 .5 . Corollary . Suppose G acts freely on X. Then a G-map f : X -> X is equivariantly deformable to a fized point free map if and only if the Lefschetz number L(f) = 0. This last corollary in the case dim X = 2 is particulary simple . Here X must be equivariantly diffeomorphic to S2 , with the antipodal action of Z 2 . An

A . VIDAL

120

equivariant map f with L(f) = 0 has degree -1 and is equivariantly homotopic to the antipodal map (cf. [9, p. 212]) . We can also mention here the corollary 6.36 in [6] . There Fadell gives, in a particular case, an obstruction theory proof of Corollary 2.5. He assumes in addition that all maps g : X --> X are homotopic and proves that the induced map f IG : XIG -> XIG can be deformed to a fixed point free map . Consequently, the map f is equivariantly homotopic to a fixed point free map . We have also proved this last fact, as in the case of the antipodal action of 7[2 on S", where all induced maps on R P 2 " have fixpoints . 2 .6. Remark. The condition codim XG >_ 3 in the Corollary 2 .4 is only necessary in our proof of Theorem 2.3 (see comments before Lemma 2.2) . However, our Corollary also holds without restrictions on the codimension . To see this, it is enough to note that the space X - XG IG of orbits is connected and consequently the Wilczynski invariant (cf . [12, p. 50]) vanishes if we assume that the Lefschetz numbers L(fG ) and L(f) are zero. We denote by G-simply connected manifold a G-manifold X such that XH is 0 and 1-connected for all subgroups H of G. Problem. Let f : X -> X be a G-map, where X is a G-simply con-

nected manifold. Suppose that the equivariant fixed point índex of f is zero or equivalenty, L(fH ) = 0 for all subgroups H of G . ft is then true that if f is equivariantly deformable to a fixed point free map?

Notice that what we need to prove is that the Wilczynski invariant of f is zero . For free and semifree actions the answer of this question is yes . But I do not know the answer in the general case. A negative answer would provide an example of an equivariant property,of a G-map f, Le . "equivariant deformability to a fixed point free map ", which does not necessary hold even if the maps fH : Xx -> Xx are deformable to fixed point free maps in the classical sense. References MASSEY W . S ., The homotopy group of a triad I.,

1.

BLAKERS A . L . ;

2.

BIERSTONE E ., The equivariant covering homotopy property for differentiable G fibre bundles, J. of Diff. Geom . 8 (1973) . BREDON G ., Equivariant cohomologies theories, Lecture notes in Math 34 (1967), Springer. BREDON G ., Introduction to compact transformation groups, Academic Press (1972) . FADELL E ., On the coincidence theorem of F.B. Fuller, Pac. J. of Math . 15 (1965) .

3. 4. 5.

Annals of Math . 53 (1951) .

DEFORMATION OF MAPS

6.

12 1

FADELL E . ; HUSSEINI S ., Fixed point theory for non simply . connected manifo1ds, Topology 20 (1981) . 7. ILLMAN S ., Smooth equivariant triangulations of G-manifolds for G a finite group, Math. Annalen 233 (1978) . 8. JANICH K ., Differenzierbare G-Mannigfaltigkeiten, Lecture Notes in Math. 59, Springer. 9. TOM DIECK T ., Transformation groups and representation theory, Lecture Notes in Math 766 (1979), Springer. 10 . WALL C .T .C ., On Poincaré complexes I, Annals of Math 86 (1967) . 11 . WALL C .T .C ., Surgery on compact manifolds, Academic Press (1970) . 12 . WILCZYNSKI D ., Fixed point free equivariant homotopy classes, Fund. Math. 123 (1984) .

Institut d'Estudis Catalans Centre de Recerca Matemática Apartat 50 , Bellaterra (Barcelona, SPAIN. current address: Universidad de La Laguna Facultad de Matemáticas La Laguna, Tenerife, SPAIN.

Rebut el 16 de Jung de 1987